A cone has a circular base radius 1, and vertex of height 3 directly
above the center of the circle. A cube has four vertices in the base
and four on the sloping sides. What is the length of a side of the
cube?

The volume of a tetrahedron is one-third the distance from a vertex to
the opposite face, times the area of that face. Find a formula for the
volume of a tetrahedron in terms of the coordinates of its vertices P, Q,
R, and S.